Understanding Divisibility by 9

This question asks us to find the smallest number that needs to be added to 45680 so that the resulting number is perfectly divisible by 9.

Divisibility Rule for 9 Explained

A simple way to check if a number is divisible by 9 is to look at the sum of its digits. If the sum of the digits is divisible by 9, then the original number is also divisible by 9. If the sum of the digits is not divisible by 9, the number is not divisible by 9.

To make a number divisible by 9, we need to add a value such that the new sum of digits is the next multiple of 9.

Applying the Divisibility Rule to 45680

Let's find the sum of the digits of the number 45680:

Sum of digits = $4 + 5 + 6 + 8 + 0$
Sum of digits = $23$

The sum of the digits is 23. Now, we need to find the smallest number to add to 45680 so that the new sum of digits is a multiple of 9.

The multiples of 9 are 9, 18, 27, 36, and so on.

Our current sum of digits is 23. The next multiple of 9 after 23 is 27.

Calculating the Required Addition

To get from a sum of 23 to the next multiple of 9, which is 27, we need to add:

Required addition to sum of digits = $27 - 23 = 4$

If we add 4 to the original number 45680, the new number becomes 45680 + 4 = 45684.

Verification of the Result

Let's check the sum of the digits of the new number, 45684:

Sum of digits of 45684 = $4 + 5 + 6 + 8 + 4$
Sum of digits of 45684 = $27$

Since 27 is divisible by 9 ($27 \div 9 = 3$), the number 45684 is divisible by 9.

Therefore, the number that must be added to 45680 to make it exactly divisible by 9 is 4.

Final Answer Determination

Based on our calculation using the divisibility rule of 9, adding 4 to 45680 makes the resulting number divisible by 9.

Comparing this with the given options:

• 0
• 1
• 5
• 4

The required number to add is 4, which matches option 4.

Revision Table: Key Concepts for Divisibility

Additional Information: Related Divisibility Rules

Understanding divisibility rules helps in quickly determining if a number can be divided evenly by another number without performing the actual division. Here are a few other common rules:

• Divisibility by 2: A number is divisible by 2 if its last digit is an even number (0, 2, 4, 6, or 8). Example: 45680 is divisible by 2 because its last digit is 0.
• Divisibility by 3: A number is divisible by 3 if the sum of its digits is divisible by 3. Example: Sum of digits of 45680 is 23. 23 is not divisible by 3.
• Divisibility by 5: A number is divisible by 5 if its last digit is 0 or 5. Example: 45680 is divisible by 5 because its last digit is 0.
• Divisibility by 10: A number is divisible by 10 if its last digit is 0. Example: 45680 is divisible by 10 because its last digit is 0.